Variations on the Post Correspondence Problem for Free Groups

Post's Correspondence Problem for hyperbolic and virtually nilpotent groups

“…If at least one of $$g$$ and $$h$$ is injective we get $$\ker (g)\cap \ker (h)=\lbrace 1\rbrace$$, so the PCP has the same statement as the classical one (‘is there any $$x \ne 1$$ such that $$g(x)=h(x)$$?'). However, when neither map is injective, $$\ker (g)\cap \ker (h)$$ is always non‐trivial as either $$F(\Sigma )$$ is cyclic, and here all non‐trivial subgroups have non‐trivial intersection, or $$F(\Sigma )$$ is non‐cyclic whence $$\ker (g)\cap \ker (h)$$ contains the non‐trivial subgroup $$[\ker (g), \ker (h)]$$ (see [9, Lemma 1]), and so while the non‐triviality of $$\operatorname{Eq}(g, h)$$ is established, it does not capture the core of the problem. We therefore quotient out by $$\ker (g)\cap \ker (h)$$ as we wish to consider the case when neither map is injective.…”

“…One motivation to study the PCP in hyperbolic groups comes from the fact that the PCP for free groups can be traced to Stallings in the 1980s (see [41]), and while recent results settle the PCP for certain classes of free group maps [4,[8][9][10]30], its solubility for general free group maps remains an important open question ( [15,Problem 5.1.4]).…”